[6] S. Gutman and G.I. Sivashinsky, Physica D43 (1990) 129. [7] G.I. Sivashinsky, Philos. Trans. R. Soc. A 332 (1990) 49. This research was supported in part by ...
Physics LettersA 175 (1993) 409—414 North-Holland
PHYSICS LETTERS A
Quasi-equilibrium in upward propagating flames A.B. Mikishev and G.I. Sivashinsky Raymond and Beverly Sackler Faculty ofExact Sciences, School of Mathematical Sciences, TelAviv University, Ramat-Aviv 69978, Israel and The Benjamin Levich Institute for Physico-Chemical Hydrodynamics, City College of New York, New York, NY 10031, USA Received 30 June 1992; accepted for publication 15 October 1992 Communicated by D.D. Holm
The Boussinesq-type model for flame—buoyancy interaction is considered. It is argued that the parabolic fronts occurring in upward propagating flames may actually be merely quasi-equilibrium transient states which eventually collapse to a stable configuration in which the inclined flame as it were spreads along the wall of the channel.
1. Introduction Thermal expansion of the gas accompanying flame propagation makes the latter sensitive to external acceleration. In upward propagating flames, the cold (denser) mixture is superposed over the hot (less dense) combustion products. For this reason, the plane flame front separating the cold and hot gases is subject in this case to the classical effect of Rayleigh—Taylor instability ~“. As a result, the flame front becomes convex toward the cold gas [1,2] (fig. 1). As is known from many experimental observations, upward propagating flames often assume a characteristic paraboloidal shape with the tip of the paraboloid located somewhere near the channel’s centerline (fig. 1). Flames where the tip slides along the channel’s wall have also been observed [3], however, this type of flame configuration somehow received less attention. Upward flame propagation, thus, may occur through different but seemingly stable geometrical realizations. The present study is attempted to gain a better understanding of the pertinent nonlinear phenomenology, which transpires to be rather interesting. As a mathematical model we shall employ the
g
Fig. 1. An upward propagating methane—air flame in a 5.1 cm diameter tube. Adapted from a schlieren photograph presented in ref. [1].
weakly nonlinear flame interface evolution equation proposed by Rakib and Sivashinsky [4]. Within the framework of a one-dimensional slab geometry, which will be discussed here, this equation reads
F~—
~ UbF~= DMFXX
+
—~--
(F— )
.
(1.1)
2 Ub ~ In combustion, in contradistinction to the Rayleigh—Taylor problem, the interface is permeable, since here the gas has a nonzero normal velocity relative to the flame front. Elsevier Science Publishers B.V.
Here y= F(x, front y= Ubt;
1) is the
perturbation of a planar flame 409
Volume 175, number 6
=
~F(x,
t)
PHYSICS LETTERS A
dx
is the space average over the gap between vertical walls x=0 and x=L. The walls are assumed to be thermally insulating. Hence, eq. (1) should be solved subject to the adiabatic boundary conditions F~(0,t)=F~(L,t)=0. (1.2) Ub is the flame speed relative to the burned gas; g is the acceleration of gravity; y= (p~—Pb)/PU is the thermal expansion parameter; p~,Pb are the densities of the unburned (cold) and burned (hot) gas, respectively; DM = D~h[ ~fl( Le 1) 1] is the Markstein diffusivity; Dth is the thermal diffusivity of the mixture; /3 is the Zeldovich number; Le is the Lewis number assumed to be high enough to ensure the positive sign of DM. Equation (1.1) was derived within the framework of the Boussinesq-type model for flame—buoyancy interaction which neglects density variation everywhere but in the external forcing term. The weakly nonlinear dynamics described by eq. (1.1) corresponds to the limit —
—
47t(2—y)U~/ygL>>l
(13)
26Apr11 1993
b=const, which For however unstable at many becomes experimentally typical ,exhibiting all three phases of the flame interface dynamics. At y = 0.8, Ub = 50 cm/s, g= 1000 cm/s2 and =0.Ol 15 (fig. 3), (2.3) yields L= 1.65 cm. The quasi-equilibrium state collapses at t~90which for the above choice of parameters corresponds to the time duration of ~i= 11.25 s. Here the flame manages to cover the distance of &~343L 565 cm. For moderately extended systems, which are usually used
2. Numerical simulations
in the laboratory, the quasi-equilibrium state remains effectively frozen. We wish to emphasize that
In nondimensional formulation problems (1.1), (1.2) may be written as (2.1)
the duration of the transient state strongly depends on the initial data. It may disappear entirely if, for example, the initial disturbance is located near one of the walls (e.g. I(~, 0)=0.0l~exp(—l00~2)or
(2.2)
~0)r0.0lç~exp[—100(~—l)2]).
where ~=x/L,
r=ygt/2U~,
=2DM
Ub/ygL.
(2.3)
Problem (2.1), (2.2) admits a basic planar solution, 410
Volume 175, number 6
PHYSICS LETTERS A
26 April 1993
b
a
V.
0.0
0.2
0.4
0.6
0.0
.0
0.0
15.2
30.1
45.6
60.6
76.0
2],e=0.02.
Fig. 2. Numerical simulation of the initial-boundary value problem (2.1), (2.2) at ~ 0)=0.0l (1—0.3) exp[100(~—0.3) (a) Equilibrium solution ~ z) at t= 76. (b) Temporal evolution ofthe flame speed V(T) = I
3. Some analytical considerations At smaller ~ (e.g. for =0.005) the time interval corresponding to the transient phase may become prohibitively long numerically, creating an illusion that the flame reached the final equilibrium. The mathematical nature of this curious effect may be explained as follows. Firstly, it is convenient to reformulate the prob1cm (2.1), (2.2) intermsoftheslopeh=cP~, hT—hh~=Eh~+h,
(3.1)
h(0, r)=h(l, r)=0.
(3.2)
of periodic solutions. The dotted curve corresponds to the asymmetric state similar to that plotted in fig. 3a. As one can see, it is not an equilibrium solution. The quasi-equilibrium of the parabolic solution stems from the fact that at ~
